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Pspace
Suppose $X$ is a completely regular topological space. Then $X$ is said to be a Pspace if every prime ideal in $C(X)$, the ring of continuous functions on $X$, is maximal.
For example, every space with the discrete topology is a Pspace.
Algebraically, a commutative reduced ring $R$ with $1$ such that every prime ideal is maximal is equivalent to any of the following statements:

$R$ is vonNeumann regular,

every ideal in $R$ is the intersection of prime ideals,

every ideal in $R$ is the intersection of maximal ideals,

every principal ideal is generated by an idempotent.
When $R=C(X)$, then $R$ is commutative reduced with $1$. In addition to the algebraic characterizations of $R$ above, $X$ being a Pspace is equivalent to any of the following statements:

if $f,g\in C(X)$, then $(f,g)=(f^{2}+g^{2})$.
Some properties of Pspaces:
1. Every subspace of a Pspace is a Pspace,
2. Every quotient space of a Pspace is a Pspace,
3. 4. Every Pspace has a base of clopen sets.
For more properties of Pspaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see here.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Mathematics Subject Classification
54E18 no label found16S60 no label found Forums
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